MHB Can a Natural Number Satisfy n ≡ 1 (mod p) for All Primes in a Large Set?

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The discussion centers on whether a natural number can satisfy the condition n ≡ 1 (mod p) for all primes in a large set of consecutive primes. The original claim suggests that for sufficiently large h, no such natural number n exists. However, a counterexample is provided, demonstrating that n can be expressed as the product of the primes plus one, which satisfies the condition. It is clarified that the primes do not need to be consecutive for this to hold true. The conclusion is that the initial claim is incorrect.
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Let $\left\{ p_{1},p_{2},\dots,p_{h}\right\}$ a set of consecutive prime numbers. I want to show that, if $h$ is large enough, then doesn't exists a natural number $n$ such that $$n\equiv1\textrm{ mod }p_{i},\,\forall i=1,\dots,h.$$
I think is true but I have no idea how to prove it. Am I wrong?
 
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Your claim is unfortunately false. Try $n = p_1 p_2 \cdots p_h + 1$ (and the primes don't need to be consecutive either).
 

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